conditional statements and...

Conditional statements and arguments

Phil 12: Logic and Decision MakingFall 2010

UC San Diego10/1/2010

Friday, October 1, 2010

Checking your progress on Inquiry

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Checking your progress on Inquiry

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Review

• Conditional statements (IF A, THEN B)

• Truth conditions:

- false when antecedent is T, consequent is F

- true otherwise

• IF A, THEN B = IF NOT-B, THEN NOT-A.


Conditional statements: IF, THEN vs. ONLY IF

• Compare:

- If you trespass, then you will be arrested

- Only if you trespass will you be arrested

• B only if A is equivalent to If B, then A.

• What follows only if is the consequent

- If you were arrested, then you trespassed

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• False if you don’t trespass and are arrested

• False if you trespass and are not arrested


Conditional statements: unless

Unless can also be used to assert conditional relations

Rule: Unless = If not

Unless you complete the assignment, you will not get promoted

says the same thing as

If you do not complete the assignment, then you will not get promoted

and

If you get promoted, then you completed the assignment.

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Clicker question 1

Which statement is not equivalent to the others?

A. “If there is a storm, the dogs will bark”

B. “Only if the dogs bark is there a storm”

C. “Only if there is a storm will the dogs bark”

D.They’re all equivalent

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Clicker question 2Assume:

What is the truth value of the statement:

If we do not need to build a new plant, then our sales are not increasing.

A.True

B. False

Sales are increasing = T Our sales force is less effective = F

We need to build a new plant = F We have excess production capacity = T

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Only if sales are increasing do we need to build a new plant.

A.True

B. False



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Unless we have excess production capacity, we do not need to build a new plant

A.True

B. False



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Sufficient Conditions• When a conditional statement uses general terms (e.g., dog,

mammal) it expresses relations between categories of things that satisfy those terms

If something is a dog, then it is a mammal

Presents a relation between being a dog and being a mammal

It asserts that meeting the first condition (being a dog) suffices for meeting the second condition (being a mammal)

If _________, then__________

suffices for11


Necessary Conditions• Remember that a true conditional statement cannot have

a true antecedent and a false consequent

• This means the consequent of a conditional expresses something that is necessary if the antecedent is true

If something is a dog, then it is a mammal

Asserts that meeting the second condition (being a mammal) is necessary for meeting the first condition (being a dog)

If _________, then__________

necessary for12


If vs. Only If again• What follows the if of a conditional is a sufficient condition

• What follows only if is a necessary condition

- You can vote only if you are at least 18 years old

Being 18 is a necessary condition for voting

- If you are able to vote, then you are at least 18 years old

Being able to vote is sufficient (evidence) that you are at least 18 years old

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Clicker question 5

Consider the statement:

Unless you go to medical school, your parents won't be happy.

You go to medical school is:

A. a necessary condition for your parents' being happy

B. a sufficient condition for your parents' being happy

C. neither necessary nor sufficient for your parents' being happy

D. both necessary and sufficient condition for your parents' being happy

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The statement:

Taking the SAT is a sufficient condition for being admitted to UCSD

is false if:

A. Someone takes the SAT and is admitted to UCSD

B. Someone takes the SAT and is not admitted to UCSD

C. Someone does not take the SAT and is admitted to UCSD

D. Someone does not take the SAT and is not admitted to UCSD

Clicker question 6

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Using conditionals in argumentsThere are two ways to use a conditional statement in a valid inference:

If A, then BA ∴B Modus ponens

• The first, obvious way:

If A, then BA B

- Why?• If B weren’t true, and A is true, then If A, then B

would be rendered false

• So the following argument form is valid:


Using conditionals in arguments

• The second, less obvious way:

From If A, then B, what happens if B is denied?

If B is false and A is true, then what is the truth value of If A, then B?

It is false. Thus A cannot be true when the whole conditional is true; we know Not A is true.

There are two ways to use a conditional statement in a valid inference:

If A, then BNot B ∴Not A Modus tollens

• So the following argument form is valid:


Uses of conditional arguments inscientific reasoning

• Modus ponens is most commonly invoked to make predictions from a hypothesis

If malaria is transmitted by mosquitoes and we eliminate the mosquitoes, then malaria will decline.

Malaria is transmitted by mosquitoes and we are eliminating the mosquitoes.

∴ Malaria will decline.

• Modus tollens is most commonly invoked to confirm or falsify a hypothesis based on the truth of falsity of a prediction


Invalid conditional arguments• What can you conclude (validly) from:

If A, then BNot A ?

• Remember, to be valid, it must be that if the premises were true, the conclusion would also have to be true

• What can we know about the truth of the conclusion?

- Both B and not B are compatible with the premises

- There is no valid argument here!

Denying the AntecedentINVALIDX


Invalid conditional arguments

• What about this:

If A, then BB ∴ A

• If the premises were true, must the conclusion be true in this case?

- No. Both A and Not A are compatible with these premises

- There is no valid argument here either!

Affirming the ConsequentINVALIDX


Summary: Conditional Arguments

• Valid argument forms:

If A, then B If A, then BA Not B ∴B ∴Not AModus ponens Modus tollens

• Invalid argument forms:

If A, then B If A, then BNot A B ∴Not B ∴ADenying the antecedent Affirming the consequent


Clicker question 7

• What form does this argument have?

I know I passed since I took the test, and if I took the test, I passed.

A. Modus ponens

B. Affirming the consequent

C. Modus tollens

D. Denying the antecedent


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